1. How well do they work? Do they display a picture properly?
Yes, depending of the type. The internal ones (usually soldered) produce a more stable picture. I use an external flicker (Apollo) and I find it very good. The quality is "a little" bit worser than a standard VGA PC but 10 times better than the TV set or the 1048 specially using 640x512 interlaced workbench setups.
2. Would I have any problems playing older games or demos with one? How is the compatibility in general?
Scan doublers are not video cards. They just take the standard amiga signal and double the refresh rate. Old games *do work* ;-)
3. NTSC and PAL video modes: Once I use a scan doubler does this issue go away Or am I still gonna have PAL software not display correctly?
If it's an issue using a 1048 CRT, it will ALSO be an issue using the Scan Doubler/FF. If a part of the screen seems eaten because you are running the Amiga in NTSC mode and the game was meant for PAL, there's no way around it, just set the machine to PAL everytime you want to run a PAL app/game.
4. Any recommended one to buy?
I have an Apollo which is supposed to be crappy but I love it, specially because I paid 100 U$S (new) and is external -works with every type of Amiga.